Chalmers University of Technology
Runaway positrons in magnetized plasmas
Tünde Fülöp and Gergely Papp
Department of Applied Physics
Fusion Theory
Chalmers University of Technology
Positrons in tokamak plasmas
• Large quantities of positrons are produced in
fusion plasmas.
ITER
– What is the positron production rate in the
presence of runaway electron avalanching?
– What is the fate of the positrons (lifetime,
runaway fraction)?
– Can we detect them?
• Main conclusion:
– Positrons in tokamaks are produced with
high energies (1 ≪ γ+ ≪ γe ).
– Most of them run away and live for several
seconds.
– Detection of synchrotron radiation may be
possible.
Under construction in France. First plasma in 2020.
Fusion Theory
Chalmers University of Technology
The friction force
• Massive runaway if E > ED (vc = vT )
• Electrons are accelerated by an electric
field and experience friction from
collisions.
• For a given electric field there is a critical
velocity, vc , at which the friction equals
the electric force.
• Runaway acceleration of some electrons
if E > Ec
Friction force
• Friction is non-monotonic function of
velocity.
ED
ne e3 ln Λ
=
4πǫ20 T
eED
eE
eE
c
0
0 ∼ Te
1/2 mev2c
Te
Ec
=
≪1
2
ED
me c
Fusion Theory
∼ mec2
Energy
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Primary and secondary generation of runaways
• Thermal electrons diffuse
through the tail of the distribution.
• Results in a small runaway population.
• In a close Coulomb collision
an existing runaway electron
can throw a thermal electron
above the runaway threshold.
• Exponential growth
of runaways.
Fusion Theory
Chalmers University of Technology
Runaway avalanches
• In tokamak disruptions:
– the plasma cools quickly,
– the resistivity η ∝ T −3/2 rises, and
– a high electric field is induced to
maintain the plasma current.
• If Ek > Ec runaway electrons are created.
• The pre-disruption current is partly
replaced by a current of runaway electrons
(Ir ≃ 1 MA in medium-sized tokamaks).
• Typical runaway electron energy: 10-20
MeV.
Fusion Theory
Carbon dust particles produced when runaways hit
a plasma-facing component in Tore Supra.
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Electron-positron pair production
• Estimated number of positrons in tokamaks with Ir = 1 MA: Np ≃ 1014
[Helander & Ward, PRL (2003)]
• This only takes into account collisions with hydrogenic ions, so it is probably an
underestimate.
• Number increases in tokamaks with larger currents (such as ITER).
Fusion Theory
Chalmers University of Technology
Runaway distribution function
• Distribution of secondary runaways:
feRE =
Êp2e⊥
pek
nr Ê
exp −
−
2πcz pek ln Λ
cz ln Λ
2pek
!
,
15
x 10
14
[Rosenbluth & Putvinski, NF 37 (1997)]
12
10
−3
f (m )
where
dnr /dt = (E − 1)nr /cz τ
8
6
4
E = e|Ek |τ /me c,
τ is the collision time for relativistic electrons,
Ê = (E − 1)/(1 + Zeff ),
p
cz = 3(Zeff + 5)/π
p = γv/c normalized momentum
Fusion Theory
0
2
0.2
0
0.4
25
0.6
30
0.8
35
p||
1
p⊥
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Positron production cross section
• Cross section for pair production (blue)
γ
+
x
e
0
σtot = aZ 2 ln3
3 + x0
Σ@ Μb D
1000
10
Σ tot
Σ Γ p1
0.1
Σ th
Σ an
10
15 20
Γ
30
50
70 100
a = 5.22 µb, x0 = 3.6
[D. A. Gryaznykh, Phys of Atomic Nuclei 61 (1998)]
• High-energy limit (red)
σγ≫1
28(Zαre )2 3
≃
ln γe
27π
• Threshold limit (green)
σth = 0.013Z 2 (γe − 3)3 µb
• Annihilation cross section (purple)
Fusion Theory
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Positron production rate
with Σ tot , ln L=10
dS p  d p e
1.5
with Σ Γ p1 , ln L=10
with Σ tot , ln L=15
1.0
with Σ Γ p1 , ln L=15
• The positron production rate
R df+ 3
dn+ /dt =
d p+ ≡ Sp is
dt
Sp = ni
Z∞
feRE σtot ve d3 pe ,
pmin
0.5
ni is the number density of the ions.
0.0
20
40
60
pe
80
100
• To take into account collisions with impurities
and electrons this should be multiplied with
P
Mp ≡ 1 + ne /ni + z nz Z 2 /ni .
• 1 g of carbon distributed uniformly in a
volume of about 80 m3 , would correspond to a
multiplicative factor of Mp ≃ 450.
• Number of positrons should be around 1015 .
Fusion Theory
Chalmers University of Technology
Energy losses
Energy loss mechanisms
8
10
dE/dt [eV/s]
7
10
6
10
5
10
Collisional
Bremsstrahlung
Synchrotron
v /v = 0.1
⊥
||
4
10
0
10
Fusion Theory
20
γ
30
40
50
Chalmers University of Technology
Steady-state positron distribution
∂f+
1 ∂ 2
= 2
(1 + p+ )f+ − ne v+ σa f+ + sp (p+ )
∂t
τ p+ ∂p+
R RE
where sp ≡ df+ /dt = ni fe σtot ve F(pe , p+ )d3 pe and F(pe , p+ ), is the probability
distribution of positrons of momentum p+ generated from electrons of energy pe .
• Most positrons that survive the slowing
down without annihilation will have
momentum below p+ = 10.
0.004
f + n +
0.003
• Here, the presence of the electric field was
neglected. But if E > Ec a population of
runaway positrons may be formed.
0.002
0.001
0.000
5
10
p+
Fusion Theory
15
20
Chalmers University of Technology
Runaway positrons
• The number of positrons that run away
can be estimated by calculating how many
positrons are born above the critical
√
velocity vc = c/ 2E.
R∞ + 2
+
• nrun = 4π pc f (p − p2c )dp
+
n run
n+
0.98
0.96
0.94
0.92
0.90
1.0
1.5
2.0
3.0
E
Fusion Theory
5.0
7.0
10.0
• For most positrons pc ≪ p+ → almost
the whole positron population can be
expected to run away n+
run ≃ n+ .
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Lifetime and possibility for detection
• The lifetime of a positron can be
estimated from the annihilation cross
section τp = 1/ne v+ σa .
16
14
12
Τp @sD
10
• Annihilation radiation is hard to detect
because the Bremsstrahlung from the
runaway electron population is larger.
8
6
4
0
5
10
p+
Fusion Theory
15
20
• Bremsstrahlung and synchrotron radiation
from runaway positrons is peaked in the
direction opposite from that of the
runaway electrons.
Chalmers University of Technology
Synchrotron radiation
3
3
10
Zeff= 1.6
P [10−9 W/µm] / n+ [m−3]
P [10−9 W/µm] / n+ [m−3]
10
Zeff= 5.0
2
10
Zeff= 10.0
1
10
0
10
0
lnΛ= 8
lnΛ= 12
lnΛ= 16
2
10
1
10
0
5
10
λ [µm]
Effective charge
Fusion Theory
15
20
10
0
5
10
λ [µm]
Coulomb logarithm
15
20
(e)
Chalmers University of Technology
Positrons in thunderstorms
• Runaway electron avalanches play a role
in lightning initiation.
(Gurevich et al, Phys. Lett. A, 1992)
• Runaway avalanching provides
mechanism for breakdown at 2 kV/cm,
rather than the conventional threshold for
breakdown in air 23 kV/cm.
• Positrons are generated and they produce
new runaway electrons.
• Feedback of positrons have been shown to
be important in lightning initiation (in
simulations).
Monte Carlo simulations showing runaway avalanches for air with
electric field E = 750 kV/m. Blue trajectories are positrons.
[From Dwyer, Phys. Plasmas (2007).]
Fusion Theory
Chalmers University of Technology
Conclusions
• Runaway avalanching is an issue of great concern in tokamaks. There are still many
unsolved questions regarding generation, propagation and losses of runaway electrons.
• Positrons in tokamaks are produced with high energies (1 ≪ γ+ ≃ γe ). Most of them run
away and live long.
• There should be more than 1015 positrons in a typical tokamak disruption with runaway
avalanching.
• Detection of synchrotron radiation (with wavelengths of a few µm) from runaway positrons
may be possible.
• Since the radiated power and spectrum shape are sensitive to the impurity concentration,
temperature and other parameters, positron radiation can be a diagnostic tool to understand
the properties of the discharge.
Fusion Theory